Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Recognize side-angle-side data.
- Apply the cosine correction.
- Find triangular area using sine.
Before you start
Squares, square roots, and sine/cosine of 60°.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Two straight paths from one point are 6 m and 8 m long, separated by a 60° angle. Find the distance between their far ends and the enclosed triangle's area.
Why this math matters
The Pythagorean theorem handles a right angle. The cosine correction extends that relationship to other included angles, letting a diagram's opening angle affect its far-end separation.
Set up the model
A useful answer starts with clear assumptions:
- Paths are straight segments in a plane.
- The 60° angle is between the given 6 m and 8 m sides.
- Only geometric distance and area are requested.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Find the gap between two angled paths
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Two straight paths from one point are 6 m and 8 m long, separated by a 60° angle. Find the distance between their far ends and the enclosed triangle's area.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Compute the squared separation
c² = 6² + 8² − 2(6)(8)cos60° = 100 − 48 = 52
The angle lies between the two multiplied sides. Since its cosine is positive, the third side is shorter than for a right-angle opening.
Recover the distance
c = √52 = 2√13 ≈ 7.211 m
The equation found a squared length. Take its positive square root to obtain a physical distance.
Find the triangle's area
A = (1/2)(6)(8)sin60° = 12√3 ≈ 20.785 m²
Sine converts one sloping side into perpendicular height. This explains why both side lengths alone cannot determine the area.
The result
The far-end distance is about 7.21 m and the area is about 20.78 m².
Opening the angle toward 90° increases both separation and area here. Beyond 90°, separation continues increasing while the sine-based area starts decreasing.
Common mistakes to catch
- Leaving out the cosine term incorrectly assumes a right angle.
- The angle must be included between the specified sides.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Keep sides 6 and 8 but use an included angle of 90°. Find the third side.
Show a hint
cos90° is zero.
Reveal answer and explanation
10
The formula reduces to √(36 + 64) = 10.
Practice 2
Sides 5 and 7 enclose angle C opposite side 8. Find cos C.
Show a hint
Rearrange the law of cosines.
Reveal answer and explanation
1/7
cos C = (25 + 49 − 64)/(2 × 5 × 7) = 10/70.
Take the idea with you
Choose the law of cosines for two sides with their included angle, or for three known sides.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Read the timing and height of a repeating motion
Completion is your own study record, not a test score. It stays in this browser, does not sync to another device, and can be removed by clearing browser data.